2 BLUEs for Default Probabilities Konstantin Vogl and Robert Wania Technische Universität Dresden, Germany Department of Business Management and Economics This Version: December 2, 2003 Abstract Assigning default probabilities to rating classes is a prerequisite for most credit risk models. In this paper the best linear unbiased estimator for the default probabilities is obtained as Aitken-estimator for correlated defaults. The estimation requires only a minimum of assumptions. Remarkable in these findings is that the Aitken-estimator uses the number of defaults in one rating class to estimate the default probability of another rating class, provided they are correlated. 1 Introduction Default probabilities play an essential role in credit risk models. The estimation of default probabilities is therefore a crucial problem. There are different estimation methods applied in practice which usually require extensive assumptions. Maximum-Likelihood based methods need distributional assumptions, scoring systems rely on explaining variables, or other methods simply assume independence of defaults to use well known estimators. In this paper only a minimum of assumptions is made for the estimation procedure. A prerequisite is that obligors are grouped together in rating classes. The estimation is based on historical default data which is available for all rating classes. This makes the method valuable especially for the retail banking portfolio where other methods fail which rely on bond spreads or balance sheet data. To achieve sound estimates, a generalized least squares estimator is used. This estimator is the best linear unbiased estimator (BLUE). In the following sections the variables and assumptions are described, the estimator is set up and special cases for applications are discussed. 2 The Framework In this section the basic variables and model assumptions are introduced. For the estimation problem historical default data of time periods t = 1,...,T is used. Within each period the rating classes r = 1,...,R are considered. In period t 1

3 and rating class r there are n tr obligors. The default variable if obligor i in rating class r 1 defaults over time period t, Y tri := i =1,...,n tr, 0 otherwise, is assigned to the i-th obligor of rating class r in time period t. Similar obligors are grouped together in each rating class, therefore these obligors can be assumed to have the same default probability. This homogeneity leads to E(Y tri )=:π r, under the assumption that the default probability of each rating class is constant over time. The Bernoulli-distributed default variables have the variance σ 2 r := V(Y tri )=π r (1 π r ). Additionally to the time stability of the default probabilities, it is assumed that the covariance of the default variables is constant over time 3 The Estimator Cov(Y tri,y tsj )=:γ rs for i j if r = s. For the estimation it is sufficient to consider the relative default frequencies K tr := 1 n tr n tr The relative default frequencies for different rating classes are combined in a vector for each time period i=1 Y tri. K t := (K t1,...,k tr ). The relative default frequency is a straightforward unbiased estimator for the unknown default probabilities E(K t )=π, where π =(π 1,...,π R ). In the notation of linear regression the residuals are defined by e t := K t π, which yields K t = π + e t, with E(e t )=0. 2

5 4 Discussing the Estimator In the following the Aitken-estimator from (4) is discussed in some special cases. Single Class, Multiple Periods If there is only one single rating class r with historical data of several time periods t =1,...,T the Aitken-estimator simplifies to 3 ˆπ blue (r) := T t=1 T n tr 1+(n tr 1)ϱ rr n sr K tr, (5) s=1 1+(n sr 1)ϱ rr where ϱ rr denotes the default correlation between the default variables. Proofof(5) Corr(Y tri,y trj )=ϱ rr := γ rr, for i j σr 2 In the single class case the covariance-matrix Σ from (3) reduces to ( 1 Diag(V(K 1r ),...,V(K Tr )) = γ rr I T T +(σr 2 γ rr )Diag,..., n 1r and with M = (1,...,1) of dimension T and K =(K 1r,...,K Tr ) the equation (5) is obtained from (4). It can be remarked that even though independence of time periods is assumed, the estimator 1 T T t=1 K tr for π r is not optimal in case n tr varies over time. Multiple Classes, Single Period Considering a single time period t and R rating classes, the best linear unbiased estimator for π is simply given by 1 n Tr ˆπ blue (t) = K t, (6) the vector of the R relative default frequencies in time t. This follows from (4) with M = I R R and K = K t. 3 Please note, the estimator ˆπ (r) blue in the single class case is not the r-th element ˆπ r blue vector ˆπ blue from (4), where all R classes are considered. ) of the 4

6 5 Conclusion The Aitken-estimator provides an approach to the problem of estimating default probabilities for rating classes. Though accounting for default correlation the estimator presented needs only the assumption of time-stable default probabilities and covariances. Proposed for the case of multiple rating classes and multiple time periods, the estimator can easily be customized for various data. Applying the theoretical findings on real data shows that the Aitken-estimator can apply negative weights on observed relative default frequencies. Further, the observed relative default frequencies from all rating classes and all time periods contribute to the estimate of the default probability of each specific rating class. References [Ait35] A.C. Aitken, On least squares and linear combination of observations, Proceedings of the Royal Society of Edinburgh A55(1935), 42. [JGH + 85] G.G. Judge, W.E. Griffiths, R.C. Hill, H. Lütkepohl, and T.-C. Lee, The theory and practice of econometrics, 2 ed., John Wiley & Sons, New York, [SO91] A. Stuart and J.K. Ord, Kendall s advanced theory of statistics, 5ed., vol. 2, Edward Arnold, London,

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